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arXiv · 2307.03624

Rigidity of min-max minimal disks in $3$-balls with non-negative Ricci curvature

Laurent Mazet·Abraão Mendes
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Abstract

In this paper we prove a rigidity statement for free boundary minimal surfaces produced via min-max methods. More precisely, we prove that for any Riemannian metric $g$ on the 3-ball $B$ with non-negative Ricci curvature and $\mathrm{II}_{\partial B}\ge g_{|\partial B}$, there exists a free boundary minimal disk $Δ$ of least area among all free boundary minimal disks in $(B,g)$. Moreover, the area of any such $Δ$ equals to the width of $(B,g)$, $Δ$ has index one, and the length of $\partialΔ$ is bounded from above by $2π$. Furthermore, the length of $\partialΔ$ equals to $2π$ if and only if $(B,g)$ is isometric to the Euclidean unit ball. This is related to a rigidity result obtained by F.C. Marques and A. Neves in the closed case. The proof uses a rigidity statement concerning half-balls with non-negative Ricci curvature which is true in any dimension.

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Laurent Mazet, Abraão Mendes. 2023-07-10. Rigidity of min-max minimal disks in $3$-balls with non-negative Ricci curvature. https://arxiv.org/abs/2307.03624

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